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Algebraic Numbers with Elements of Small Height Haydar Göral Algebraic Numbers with Elements of Small Height Haydar Göral To cite this version: Haydar Göral. Algebraic Numbers with Elements of Small Height. 2017. hal-01191860v3 HAL Id: hal-01191860 https://hal.archives-ouvertes.fr/hal-01191860v3 Preprint submitted on 22 Oct 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ALGEBRAIC NUMBERS WITH ELEMENTS OF SMALL HEIGHT HAYDAR GORAL¨ Abstract. In this paper, we study the field of algebraic numbers with a set of elements of small height treated as a predicate. We prove that such structures are not simple and have the independence property. A real algebraic integer α > 1 is called a Salem number if α and 1/α are Galois conjugate and all other Galois conjugates of α lie on the unit circle. It is not known whether 1 is a limit point of Salem numbers. We relate the simplicity of a certain pair with Lehmer's conjecture and obtain a model-theoretic characterization of Lehmer's conjecture for Salem numbers. 1. Introduction The logarithmic height function measures the arithmetic complexity of an algebraic number and it plays a fundamental role in algebra and diophantine geometry. Before defining the logarithmic height function, we first define the Mahler measure of a polynomial over C: For a non-zero polynomial f(x) = ad(X − α1) ··· (X − αd) 2 C[X]; its Mahler measure is defined as the finite product Y m(f) = jadj jαjj: jαj |≥1 By convention, the Mahler measure of zero is defined to be 1. Let Q be the field of algebraic numbers. For a non-zero α in Q with irreducible polynomial f(x) 2 Z[X], we define its Mahler measure as m(α) = m(f): The absolute non-logarithmic height of α is defined by H(α) = m(α)1=d where d = deg f: Then the logarithmic height of α is defined as log m(α) h(α) = log H(α) = : d A central question in diophantine geometry is Lehmer's conjecture, which is still open and it states that there exists an absolute constant c > 1 such that if m(α) > 1 then m(α) ≥ c. This question was posed by D. Lehmer [14] around 1933, who claimed that the polynomial X10 + X9 − X7 − X6 − X5 − X4 − X3 + X + 1 has the smallest Mahler measure, which is approximately 1:17628 and this is still the smallest known Mahler measure of an algebraic number. A real algebraic integer α > 1 is called a Salem number if α and 1/α are Galois conjugate and all other Galois conjugates of α lie on the unit circle. It is not known whether 1 is a limit point of Salem numbers. This is a special case of Lehmer's conjecture since for a Salem number α we have m(α) = α, and Lehmer's conjecture for Salem numbers states that 1 is not a limit point of Salem numbers. For a detailed exposition on Mahler measure and Lehmer's problem, see [20]. Classification theory is an extensive project in model theory, emerged in the work of M. Morley in the 1960's and S. Shelah in the 1970's, pursuing the categorization of first-order theories. The question of how many models a theory can have has been at the center of model 1991 Mathematics Subject Classification. 12L12, 11G50, 03C45. Key words and phrases. height, model theory, salem numbers. Partially supported by ValCoMo (ANR-13-BS01-0006) and MALOA (PITN-GA-2009-238381). 1 2 HAYDAR GORAL¨ theory. Stable theories are important to classify their models. If a theory is not stable then its models are difficult to handle and various to classify, and arithmetic fits in this case. A theory T is said to be stable if there is no first-order formula in T which has the order property. Stable theories are equipped with a notion of independence, namely the forking independence, with a well-defined set of properties, reminiscent of the algebraic independence in fields and the linear independence in vector spaces. For instance, the theory of algebraically closed fields is stable. Conversely, stability also yields some structural properties of the models and a well- known result is that a field is !-stable if and only if it is algebraically closed [15]. A theory is simple, if the forking independence is symmetric and it is known that stable theories are simple. There are many fruitful interactions between model theory and algebra. The main examples are Ax{Grothendieck theorem via compactness [1] and Chevalley's constructibility theorem via quantifier elimination. More recently, other two important examples are the group configura- tion theorem by Hrushovski [12], which is a generalization of Weil's group chunk theorem, and Hrushovski's proof of the Mordell-Lang conjecture for function fields in all characteristics [13]. In this paper, we also study the interaction between model theory and diophantine geometry via the set of algebraic numbers with elements of small height. The results in this article are from the PhD thesis of the author [8]. The model theoretic properties of Q are well understood: it is strongly minimal and has quantifier elimination in the language of rings Lr = f+; −; ·; 0; 1g. In this paper, we mainly focus on the pair Q together with a predicate, in particular our predicate will be the elements of height less than a given positive real number. Moreover, we study these pairs in terms of stability theory. The first motivation came from B. Zilber. In 1990, B. Zilber [24] showed that the pair (C; µ) ≡ (Q; µ) is !-stable where µ is the group of complex roots of unity, and his pattern was based on the result of H. Mann [16]. The result of B. Zilber [24] was generalized by L. van den Dries and A. G¨unaydın [6] to algebraically closed fields with a multiplicative subgroup satisfying a certain diophantine condition. In this paper our predicate, namely the elements of small height, con- tains the group µ. Moreover, the theory of Q with this predicate will be a proper expansion of the theory of (Q; µ): The model theory of pairs has attracted a lot of attention recently. More generally, stable theories with a predicate were analyzed in the paper of E. Casanovas and M. Ziegler [3], where they gave criteria for a pair to be stable. Their result in [3] implies the result of B. Zilber [24] and also B. Poizat's result on the !-stability of the theory of pairs of algebraically closed fields [18]. Now we state Kronecker's theorem from diophantine geometry with no proof, as the details can be found in the book [2, 1.5.9]. Theorem 1.1. (Kronecker) Let α in Q be a non-zero algebraic number. Then h(α) = 0 if and only if it is a root of unity. Combining Zilber's result [24] with Kronecker's theorem, we see that the pair (Q; fa 2 Q : h(a) = 0g) is !-stable. Throughout this paper, the language Lm will denote the language f1; ·} where the binary operation · is the usual multiplication. Let S be the set of all Salem numbers. We put × Pb := fa 2 Q : m(a) ≤ bg and Sb = Pb \ S where b ≥ 1, and we set S" = fa 2 Q : h(a) ≤ "g where " > 0: Both Pb and S" contain the group of roots of unity µ. In other words, Lehmer's conjecture and its version for Salem ELEMENTS OF SMALL HEIGHT 3 numbers state that there exists b > 1 such that Pb = P1 = µ and Sb = S1 = ; respectively. The pairs (Q; Sb) and (Q;S") can be seen as an Lr(U) = Lr [ fUg structures where U is a unary relation symbol whose interpretations are Sb and S" respectively. We call S" the set of algebraic numbers of small height. Note that the Mahler measure of an algebraic number is again an algebraic number. Put also S = fa 2 Q : h(a) < 1g = S1; as there is no algebraic number whose height is 1 by Lindemann's theorem. Let T" be the theory T h(Q;S"): We focus on the model theory of (Q;S") in the language Lm(U) = Lm [fUg and we prove a result which shows that small perturbations of the property of being a root of unity changes drastically the stability properties of the ambient structure. This means that T" is model-theoretically very complicated. We also relate the simplicity of a certain pair with Lehmer's conjecture and give a model-theoretic characterization of a special case of Lehmer's conjecture. We prove the following theorem (see Theorem (3.4) and Theorem (4.2)). Main Theorem. The theory of (Q;S") is not simple and has the independence property in the language Lm(U). Moreover, the theory of (Q; Sb) is simple for some b > 1 in Lm(U) if and only if Lehmer's conjecture holds for all Salem numbers. Furthermore, the simplicity of the pair (Q;Pb) for some b > 1 in Lm(U) implies that Lehmer's conjecture holds for all Salem numbers. 2. Preliminaries and Height Lemmas 2.1. The Logarithmic Height Function.
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